Black hole evaporation 


A brief analysis of the mathematical results of Hawking radiation.  


Black hole evaporation.  
The Hawking temperature T of a Schwarzschild (nonrotating, uncharged) black hole with mass m is given by the equation (in geometrized units) [reference 1]  
T = hbar/(8 pi k m).  equation 1  
In conventional units (which we use here), this would be written  
T = (hbar c^{3})/(8 pi G k m).  equation 2  
The emission of this energy results in an energy decrease of the black
hole, and thus a loss in its mass. What period of time tau will it
take for a black hole of mass mu to evaporate completely? A black hole with mass m has a Schwarzschild radius 

r = 2 G m/c^{2}  equation 3  
and thus an area of  
A = 4 pi r^{2}  equation 4  
A = 16 pi G^{2} m^{2}/c^{4}.  equation 5  
Hawking radiation would have a power P related to the hole's area A and its temperature T by the blackbody power law (with e = 1),  
P = sigma A T^{4}  equation 6  
P = (sigma hbar^{4} c^{8})/(256 pi^{3} G^{2} k^{4} m^{2})  equation 7  
or more conveniently,  
P = K/m^{2}  equation 8  
where K == (sigma hbar^{4} c^{8})/(256 pi^{3} G^{2} k^{4}) = 3.563 x 10^{32} W kg^{2}. Given that the power of the Hawking radiation is the rate of energy loss of the hole, we can write  
P = dE/dt.  equation 9  
Since the total energy E of the hole is related to its mass m by Einstein's massenergy formula,  
E = m c^{2}  equation 10  
we can then rewrite P = dE/dt as  
P = (d/dt) (m c^{2})  equation 11  
P = c^{2} dm/dt.  equation 12  
We can then equate this to our above expression for the power, P = K/m^{2}, and find  
c^{2} dm/dt = K/m^{2}.  equation 13  
This differential equation is separable, and we can write  
m^{2} dm = K/c^{2} dt.  equation 14  
Integrating over m from mu (the initial mass of the hole) to zero (complete evaporation), and over t from zero to tau, we find that  
tau = c^{2}/(3 K) mu^{3}.  equation 15  
That is, the evaporation time of the hole is proportional to the cube
of its mass.


References.  
1. Black holes, white dwarfs, and neutron stars: The physics of compact objects Stuart L. Shapiro, Saul A. Teukolsky p. 366 WileyInterscience; 1983 
reference 1  
Navigation.  
Erik Max Francis
 TOP Welcome to my homepage. 


Writing
 UP Various things I've written. 


Essays
 UP Essays I've written. 


Geosynchronous and geostationary orbits
 PREVIOUS 


Gravitational redshift
 NEXT A thoughtexperiment demonstrating the existence of gravitational redshift. 


Quick links.  
Contents of Erik Max Francis' homepages
 CONTENTS Everything in my homepages. 


Feedback
 FEEDBACK How to send feedback on these pages to the author. 


About Erik Max Francis
 PERSONAL Information about me. 


Copyright
 COPYRIGHT Copyright information regarding these pages. 




Copyright © 1995 Erik Max Francis. All rights reserved. 



The Alcyone Systems Web Ring (13 sites) >>> 

[ Alcyone Systems  Erik Max Francis  Blackgirl International  Bosskey.net  CatCam  Crank dot Net  Hard Science  Losers dot Org  Polly Wanna Cracka?  Realpolitik  sade deluxe  7 Sisters Productions ] 